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Mirrors > Home > MPE Home > Th. List > lmiclcl | Structured version Visualization version GIF version |
Description: Isomorphism implies the left side is a module. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
Ref | Expression |
---|---|
lmiclcl | ⊢ (𝑅 ≃𝑚 𝑆 → 𝑅 ∈ LMod) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brlmic 20678 | . . 3 ⊢ (𝑅 ≃𝑚 𝑆 ↔ (𝑅 LMIso 𝑆) ≠ ∅) | |
2 | n0 4346 | . . 3 ⊢ ((𝑅 LMIso 𝑆) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝑅 LMIso 𝑆)) | |
3 | 1, 2 | bitri 274 | . 2 ⊢ (𝑅 ≃𝑚 𝑆 ↔ ∃𝑓 𝑓 ∈ (𝑅 LMIso 𝑆)) |
4 | lmimlmhm 20674 | . . . 4 ⊢ (𝑓 ∈ (𝑅 LMIso 𝑆) → 𝑓 ∈ (𝑅 LMHom 𝑆)) | |
5 | lmhmlmod1 20643 | . . . 4 ⊢ (𝑓 ∈ (𝑅 LMHom 𝑆) → 𝑅 ∈ LMod) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝑓 ∈ (𝑅 LMIso 𝑆) → 𝑅 ∈ LMod) |
7 | 6 | exlimiv 1933 | . 2 ⊢ (∃𝑓 𝑓 ∈ (𝑅 LMIso 𝑆) → 𝑅 ∈ LMod) |
8 | 3, 7 | sylbi 216 | 1 ⊢ (𝑅 ≃𝑚 𝑆 → 𝑅 ∈ LMod) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wex 1781 ∈ wcel 2106 ≠ wne 2940 ∅c0 4322 class class class wbr 5148 (class class class)co 7408 LModclmod 20470 LMHom clmhm 20629 LMIso clmim 20630 ≃𝑚 clmic 20631 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7411 df-oprab 7412 df-mpo 7413 df-1st 7974 df-2nd 7975 df-1o 8465 df-lmhm 20632 df-lmim 20633 df-lmic 20634 |
This theorem is referenced by: lmisfree 21396 |
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